In the making of electrical harnesses of the type composed of a large number of discrete wires, it is desirable to bind together related groups of the wires. The binding can be done with a suitable twine, wire, plastic tie strip, tape, or other similar material. Tape, having an adhesive on one side, is becoming the preferred binding medium in the manufacture of wiring harnesses because it is effective and inexpensive to use. In an automated environment, however, proper tension of the tape during the wrapping process must be maintained to avoid stretching or breaking the tape and to ensure high quality binding of the wire bundle. Further, once the tape is cut and wrapped about a wire bundle, the free end of the supply of tape must be controlled and attached to the next binding point. If too much tension is present, when the tape is pushed into the cutting blade, the tape will stretch in the area of the cut resulting in a curled end which is very difficult to control. This problem occurs when sufficient slack is not de-reeled prior to the cutting operation. Another problem that occurs when sufficient slack is not de-reeled is the difficulty in controlling the length of the portion of the tape that is actually wrapped about the wire bundle. In order to begin the wrapping operation, the end of the tape must first be attached to the wire bundle. This is normally done by pressing the adhesive side of the tape against the side of the wire bundle. If the de-reeling forces are substantial, they may overcome the adhesive forces resulting in the tape slipping or completely releasing from the wire bundle. What is needed is a tape wrapping apparatus that will provide sufficient slack in the tape while maintaining a desired level of tension during the tape wrap operation.